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2.1 – Quadratic Functions
In this section, you will learn to
analyze graphs of quadratic functions
write quadratic functions in standard form and sketch its graphs
solve real-life problems
Definition of a Polynomial Function:
1 2 21 2 2 1 0....
is a polynomial function of with degree .
n n nn n nf x a x a x a x a x a x a
x n
Definition of a Quadratic Function:
a) Axis of symmetry: the line where the parabola is symmetric
b) Vertex: The point where the axis of symmetry intersects the parabola
2 , 0f x ax bx c a
2x
2, 2
1 2 3 4 5 6-1-2
1
2
3
4
5
-1
-2
-3
-4
-5
x
y
Definition of a Quadratic Function:
c) Upward or Downward: If the leading coefficient is positive (a>0) , the parabola opens upward.
2 , 0f x ax bx c a
1 2 3 4-1-2-3-4
1
2
3
4
-1
-2
-3
-4
x
y
22 1
2 0
f x x
Definition of a Quadratic Function:
c) Upward or Downward: If the leading coefficient is negative (a<0) , the parabola opens downward.
2 , 0f x ax bx c a
22 1
2 0
f x x
1 2 3 4-1-2-3-4
1
2
3
4
-1
-2
-3
-4
x
y
Definition of a Quadratic Function:
d) Minimum or Maximum: If the parabola opens upward, the vertex has a minimum value. If the parabola opens downward, the vertex has a maximum value.
2 , 0f x ax bx c a
1 2 3 4-1-2-3-4
1
2
3
4
-1
-2
x
y
1 2 3 4-1-2-3-4
1
2
-1
-2
-3
-4
x
y
Minimum
Maximum
Standard Form of a Quadratic Function:
a) Vertex:b) Axis of Symmetry: c) Vertex:
Therefore,
* To write an equation in standard form, you need to complete the
square.
2, 0f x a x h k a
2 2
b bh k f f h
a a
, ,2 2
b bh k f
a a
,h k
x h
Identify the vertex and axis of symmetry for
There are two methods to identify the vertex and the axis of symmetry.
Method 1:
24 2 1f x x x
2 1
2 2 4 4
bh
a
21 1 1 3
4 2 14 4 4 4
k f
1Axis of Symmetry :
4x
Method 2: Complete the Square
24 2 1f x x x
21 4 2f x x x
2 11 4
2f x x x
2 1
1 42
1
4 16
1f x x x
2
3 14
4 4f x x
1 3 1, , Axis of Symmetry:
4 4 4h k x
2
1 34
4 4f x x
Complete the Square:
212 1
2f x x x
211 2
2f x x x
211 4
2f x x x
211 4
22 4f x x x
213 2
2f x x 21
2 32
f x x
, 2, 3 Axis of Symmetry: 2h k x
Method 2: Complete the Square
22 3 1f x x x
21 2 3f x x x
2 31 2
2f x x x
2 31 2
2
9
8
9
16f x x x
2
1 32
8 4f x x
3 1 3, , Axis of Symmetry:
4 8 4h k x
2
3 12
4 8f x x
Identify the vertex and zeros to graph:
Vertex:
Zeros:
22 8 3f x x x
8
22 2
h
28 8 4 2 3
2 2x
22 2 2 8 2 3 5k f
, 2,5h k
8 2 10 4 10
4 2
Identify the vertex and zeros to graph:
Vertex: Zeros:
22 8 3f x x x
2,54 10
0.42, 3.582
1 2 3 4 5-1
1
2
3
4
5
6
-1
-2
-3
-4
-5
x
y
Find the quadratic equation in standard form:
Find the standard form of the equation of
the parabola whose vertex is 2, 3 and
passes through the point 3,5 .
Find the quadratic equation in standard form:
Vertex:Point:
, 2, 3h k , 3,5x y
2f x a x h k
25 3 2 3a
5 25 3a 8 25a
28 82 3
25 25a f x x
Real-Life Example:
A baseball is hit at a point 5 feet above the ground
at a velocity of 100 feet per second and at an angle
of 45 degrees with respect to the ground. The path
of the baseball is given by the function
0f x 2.002 5 . What is the maximum
height reached by the ball?
x x
Real-Life Example:
Since this parabolic path is opening downward, the maximum height is reached
at the vertex point. You can use the formulas for h and k to find the vertex point. Then, the maximum height is represented by the k value.
Real-Life Example:
The maximum height reached by this ball
is 130 ft.
20.002 5f x x x
20.002 5f x x x
1
2502 0.002
h
2250 0.002 250 250 5 130 .k f ft
Graph:
The maximum height reached by this ball is
130 ft.
20.002 5f x x x
50 100 150 200 250 300 350 400 450 500 550-50
25
50
75
100
125
150
-25
-50
x
y
Real-Life Example:
2
A yo yo was dropped at a point 20 inches
above the ground with a velocity of 2 inch
per second with respect to the ground.
The path of the yo yo is given by the function
4 20 . Find the minimum heif x x x
ght
the yo yo will reach.
Real-Life Example:
Since this parabolic path is opening upward,
the minimum height is reached at the vertex
point. You can use the formulas for h and k
to find the vertex point. Then, the minimum
height is represented by the k value.
4
2 .2 1
h in
2 4 20f x x x
22 2 4 2 20 16 .k f in
Real-Life Example:
The minimum height reached by the yo-yo is
16 feet.
2 4 20f x x x
1 2 3 4 5 6 7 8 9-1-2-3-4-5
4
8
12
16
20
24
28
32
36
40
-4
x
y
Real-Life Example:
The minimum height reached by the yo-yo is
16 feet.
2 4 20f x x x
1 2 3 4 5-1
4
8
12
16
20
24
x
y